Echoes in classical dynamical systems

Echoes arise when external manipulations to a system induce a reversal of its time evolution that leads to a more or less perfect recovery of the initial state. We discuss the accuracy with which a cloud of trajectories returns to the initial state in classical dynamical systems that are exposed to additive noise, and small differences in the equations of motion for forward and backward evolution. The cases of integrable and chaotic motion, and small or large noise are studied in some detail and many different dynamical laws are identified. Experimental tests in two-dimensional flows that show chaotic advection are proposed.